Elementary Principles in Statistical Mechanics/Chapter VII

by Josiah Willard Gibbs

CHAPTER VII.

FARTHER DISCUSSION OF AVERAGES IN A CANONICAL ENSEMBLE OF SYSTEMS.

Returning to the case of a canonical distribution, we have for the index of probability of configuration

$\eta _{q}={\frac {\psi _{q}-\epsilon _{q}}{\Theta }}$

(178)

as appears on comparison of formulae (142) and (161). It follows immediately from (142) that the average value in the ensemble of any quantity $u$ which depends on the configuration alone is given by the formula

${\overline {u}}=\mathop {\int \ldots \int } _{\rm {config.}}^{\rm {all}}ue^{\frac {\psi _{q}-\epsilon _{q}}{\Theta }}\Delta _{\dot {q}}{}^{\frac {1}{2}}\,dq_{1}\ldots dq_{n},$

(179)

where the integrations cover all possible configurations. The value of $\psi _{q}$ is evidently determined by the equation

$e^{-{\frac {\psi _{q}}{\Theta }}}=\mathop {\int \ldots \int } _{\rm {config.}}^{\rm {all}}e^{-{\frac {\epsilon _{q}}{\Theta }}}\Delta _{\dot {q}}{}^{\frac {1}{2}}\,dq_{1}\ldots dq_{n}.$

(180)

By differentiating the last equation we may obtain results analogous to those obtained in Chapter IV from the equation

$e^{-{\frac {\psi }{\Theta }}}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{-{\frac {\epsilon }{\Theta }}}\,dp_{1}\ldots dq_{n}.$

As the process is identical, it is sufficient to give the results:

$d\psi _{q}={\overline {\eta }}_{q}\,d\Theta -{\overline {A}}_{1}\,da_{1}-{\overline {A}}_{2}\,da_{2}-\mathrm {etc.} ,$

(181)

or, since

$\psi _{q}={\overline {\epsilon }}_{q}+\Theta {\overline {\eta }}_{q},$

(183)

and

$d\psi _{q}=d{\overline {\eta }}_{q}+{\overline {\eta }}_{q}\,d\Theta +\Theta \,d{\overline {\eta }}_{q},$

(183)

$d{\overline {\epsilon }}_{q}=-\Theta d{\overline {\eta }}_{q}-{\overline {A}}_{1}\,da_{1}-{\overline {A}}_{2}\,da_{2}-\mathrm {etc.}$

(184)

It appears from this equation that the differential relations subsisting between the average potential energy in an ensemble of systems canonically distributed, the modulus of distribution, the average index of probability of configuration, taken negatively, and the average forces exerted on external bodies, are equivalent to those enunciated by Clausius for the potential energy of a body, its temperature, a quantity which he called the disgregation, and the forces exerted on external bodies.^[1]

For the index of probability of velocity, in the case of canonical distribution, we have by comparison of (144) and (163), or of (145) and (164),

$\eta _{p}={\frac {\psi _{p}-\epsilon _{p}}{\Theta }}$

(185)

which gives

${\overline {\eta }}_{p}={\frac {\psi _{p}-{\overline {\epsilon }}_{p}}{\Theta }};$

(186)

we have also

${\overline {\epsilon }}_{p}={\tfrac {1}{2}}n\Theta ,$

(187)

and by (140),

$\psi _{p}=-{\tfrac {1}{2}}n\Theta \log(2\pi \Theta ).$

(188)

From these equations we get by differentiation

$d\psi _{p}={\overline {\eta }}_{p}\,d\Theta ,$

(189)

and

$d{\overline {\epsilon }}_{p}=-\Theta \,d{\overline {\eta }}_{p}.$

(190)

The differential relation expressed in this equation between the average kinetic energy, the modulus, and the average index of probability of velocity, taken negatively, is identical with that given by Clausius locis citatis for the kinetic energy of a body, the temperature, and a quantity which he called the transformation-value of the kinetic energy.^[2] The relations

${\overline {\epsilon }}={\overline {\epsilon }}_{q}+{\overline {\epsilon }}_{p},\quad {\overline {\eta }}={\overline {\eta }}_{q}+{\overline {\eta }}_{p}$

are also identical with those given by Clausius for the corresponding quantities.

Equations (112) and (181) show that if $\psi$ or $\psi _{q}$ is known as function of $\Theta$ and $a_{1}$, $a_{2}$, etc., we can obtain by differentiation ${\overline {\epsilon }}$ or ${\overline {\epsilon }}_{q}$, and ${\overline {A}}_{1}$, ${\overline {A}}_{2}$ etc. as functions of the same variables. We have in fact

${\overline {\epsilon }}=\psi -\Theta {\overline {\eta }}=\psi -\Theta {\frac {d\psi }{d\Theta }}.$

(191)

${\overline {\epsilon }}_{q}=\psi _{q}-\Theta {\overline {\eta }}_{q}=\psi _{q}-\Theta {\frac {d\psi _{q}}{d\Theta }}.$

(192)

The corresponding equation relating to kinetic energy,

${\overline {\epsilon }}_{p}=\psi _{p}-\Theta \eta _{p}=\psi _{p}-\Theta {\frac {d\psi _{p}}{d\Theta }}.$

(193)

which may be obtained in the same way, may be verified by the known relations (186), (187), and (188) between the variables. We have also

${\overline {A}}_{1}=-{\frac {d\psi }{da_{1}}}=-{\frac {d\psi _{q}}{da_{1}}},$

(194)

etc., so that the average values of the external forces may be derived alike from $\psi$ or from $\psi _{q}$.

The average values of the squares or higher powers of the energies (total, potential, or kinetic) may easily be obtained by repeated differentiations of $\psi$, $\psi _{q}$, $\psi _{p}$, or ${\overline {\epsilon }}$, ${\overline {\epsilon }}_{q}$, ${\overline {\epsilon }}_{p}$, with respect to $\Theta$. By equation (108) we have

${\overline {\epsilon }}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\epsilon e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n},$

(195)

and differentiating with respect to $\Theta$,

${\frac {d{\overline {\epsilon }}}{d\Theta }}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\bigg (}{\frac {\epsilon ^{2}-\psi \epsilon }{\Theta ^{2}}}+{\frac {\epsilon }{\Theta }}{\frac {d\psi }{d\Theta }}{\bigg )}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n},$

(196)

whence, again by (108),

${\frac {d{\overline {\epsilon }}}{d\Theta }}={\frac {{\overline {\epsilon ^{2}}}-\psi {\overline {\epsilon }}}{\Theta ^{2}}}+{\frac {\overline {\epsilon }}{\Theta }}{\frac {d\psi }{d\Theta }},$

or

${\overline {\epsilon ^{2}}}=\Theta ^{2}{\frac {d{\overline {\epsilon }}}{d\Theta }}+{\overline {\epsilon }}{\bigg (}\psi -\Theta {\frac {d\psi }{d\Theta }}{\bigg )}.$

(197)

Combining this with (191),

${\overline {\epsilon ^{2}}}={\overline {\epsilon }}^{2}+\Theta ^{2}{\frac {d{\overline {\epsilon }}}{d\Theta }}={\bigg (}\psi -\Theta {\frac {d\psi }{d\Theta }}{\bigg )}^{2}-\Theta ^{3}{\frac {d^{2}\psi }{d\Theta ^{2}}}.$

(198)

In precisely the same way, from the equation

${\overline {\epsilon }}_{q}=\mathop {\int \ldots \int } _{\rm {config.}}^{\rm {all}}\epsilon _{q}e^{\frac {\psi _{q}-\epsilon _{q}}{\Theta }}\Delta _{\dot {q}}{}^{\frac {1}{2}}\,dq_{1}\ldots dq_{n},$

(199)

we may obtain

${\overline {\epsilon _{q}{}^{2}}}={\overline {\epsilon _{q}}}^{2}+\Theta ^{2}{\frac {d{\overline {\epsilon _{q}}}}{d\Theta }}={\bigg (}\psi _{q}-\Theta {\frac {d\psi _{q}}{d\Theta }}{\bigg )}^{2}-\Theta ^{3}{\frac {d^{2}\psi _{q}}{d\Theta ^{2}}}.$

(200)

In the same way also, if we confine ourselves to a particular configuration, from the equation

${\overline {\epsilon }}_{p}=\mathop {\int \ldots \int } _{\rm {veloc.}}^{\rm {all}}\epsilon _{p}e^{\frac {\psi _{p}-\epsilon _{p}}{\Theta }}\Delta _{p}{}^{\frac {1}{2}}\,dp_{1}\ldots dp_{n},$

(201)

we obtain

${\overline {\epsilon _{p}{}^{2}}}={\overline {\epsilon _{p}}}^{2}+\Theta ^{2}{\frac {d{\epsilon _{p}}}{d\Theta }}={\bigg (}\psi _{p}-\Theta {\frac {d\psi _{p}}{d\Theta }}{\bigg )}^{2}-\Theta ^{3}{\frac {d^{2}\psi _{p}}{d\Theta ^{2}}},$

(202)

which by (187) reduces to

${\overline {\epsilon _{p}{}^{2}}}=({\tfrac {1}{4}}n^{2}+{\tfrac {1}{2}}n)\Theta ^{2}.$

(203)

Since this value is independent of the configuration, we see that the average square of the kinetic energy for every configuration is the same, and therefore the same as for the whole ensemble. Hence ${\overline {\epsilon _{p}{}^{2}}}$ may be interpreted as the average either for any particular configuration, or for the whole ensemble. It will be observed that the value of this quantity is determined entirely by the modulus and the number of degrees of freedom of the system, and is in other respects independent of the nature of the system.

Of especial importance are the anomalies of the energies, or their deviations from their average values. The average value of these anomalies is of course zero. The natural measure of such anomalies is the square root of their average square. Now

${\overline {(\epsilon -{\overline {\epsilon }})^{2}}}={\overline {\epsilon ^{2}}}-{\overline {\epsilon }}^{2},$

(204)

identically. Accordingly

${\overline {(\epsilon -{\overline {\epsilon }})^{2}}}=\Theta ^{2}{\frac {d{\overline {\epsilon }}}{d\Theta }}=-\Theta ^{3}{\frac {d^{2}\psi }{d\Theta ^{2}}}.$

(205)

In like manner,

${\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{2}}}=\Theta ^{2}{\frac {d{\overline {\epsilon }}_{q}}{d\Theta }}=-\Theta ^{3}{\frac {d^{2}\psi _{q}}{d\Theta ^{2}}}.$

(206)

${\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{2}}}=\Theta ^{2}{\frac {d{\overline {\epsilon }}_{p}}{d\Theta }}=-\Theta ^{3}{\frac {d^{2}\psi _{p}}{d\Theta ^{2}}}={\tfrac {1}{2}}n\Theta ^{2}.$

(207)

Hence

${\overline {(\epsilon -{\overline {\epsilon }})^{2}}}={\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{2}}}+{\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{2}}}.$

(208)

Equation (206) shows that the value of $d{\overline {\epsilon }}_{q}/d\Theta$ can never be negative, and that the value of $d^{2}\psi _{q}/d\Theta ^{2}$ or $d{\overline {\eta }}_{q}/d\Theta$ can never be positive.^[3]

To get an idea of the order of magnitude of these quantities, we may use the average kinetic energy as a term of comparison, this quantity being independent of the arbitrary constant involved in the definition of the potential energy. Since

${\overline {\epsilon }}_{p}={\tfrac {1}{2}}n\Theta ,$

${\frac {\overline {(\epsilon _{p}-{\overline {\epsilon _{p}}})^{2}}}{{\overline {\epsilon _{p}}}^{2}}}={\frac {2}{n}},$

(209)

${\frac {\overline {(\epsilon _{q}-{\overline {\epsilon _{q}}})^{2}}}{{\overline {\epsilon _{p}}}^{2}}}={\frac {2}{n}}{\frac {d{\overline {\epsilon _{q}}}}{d{\overline {\epsilon _{p}}}}},$

(210)

${\frac {\overline {(\epsilon -{\overline {\epsilon }})^{2}}}{{\overline {\epsilon _{p}}}^{2}}}={\frac {2}{n}}{\frac {d{\overline {\epsilon }}}{d{\overline {\epsilon _{p}}}}}={\frac {2}{n}}+{\frac {2}{n}}{\frac {d{\overline {\epsilon _{q}}}}{d{\overline {\epsilon _{p}}}}}.$

(211)

These equations show that when the number of degrees of freedom of the systems is very great, the mean squares of the anomalies of the energies (total, potential, and kinetic) are very small in comparison with the mean square of the kinetic energy, unless indeed the differential coefficient $d{\overline {\epsilon }}_{q}/d{\overline {\epsilon }}_{p}$ is of the same order of magnitude as $n$. Such values of $d{\overline {\epsilon }}_{q}/d{\overline {\epsilon }}_{p}$ can only occur within intervals (${\overline {\epsilon }}_{p}{}''-{\overline {\epsilon }}_{p}{}'$) which are of the order of magnitude of $n^{-1}$ unless it be in cases in which ${\overline {\epsilon }}_{q}$ is in general of an order of magnitude higher than ${\overline {\epsilon }}_{p}$. Postponing for the moment the consideration of such cases, it will be interesting to examine more closely the case of large values of $d{\overline {\epsilon }}_{q}/d{\overline {\epsilon }}_{p}$ within narrow limits. Let us suppose that for ${\overline {\epsilon }}_{p}{}'$ and ${\overline {\epsilon }}_{p}{}''$ the value of $d{\overline {\epsilon }}_{q}/d{\overline {\epsilon }}_{p}$ is of the order of magnitude of unity, but between these values of ${\overline {\epsilon }}_{p}$ very great values of the differential coefficient occur. Then in the ensemble having modulus $\Theta ''$ and average energies ${\overline {\epsilon }}_{p}{}''$ and ${\overline {\epsilon }}_{q}{}''$, values of $\epsilon _{q}$ sensibly greater than ${\overline {\epsilon }}_{q}{}''$ will be so rare that we may call them practically negligible. They will be still more rare in an ensemble of less modulus. For if we differentiate the equation

$\eta _{q}={\frac {\psi _{q}-\epsilon _{q}}{\Theta }}$

regarding $\epsilon _{q}$ as constant, but $\Theta$ and therefore $\psi _{q}$ as variable, we get

${\bigg (}{\frac {d\eta _{q}}{d\Theta }}{\bigg )}_{\epsilon _{q}}={\frac {1}{\Theta }}{\frac {d\psi _{q}}{d\Theta }}-{\frac {\psi _{q}-\epsilon _{q}}{\Theta }},$

(212)

whence by (192)

${\bigg (}{\frac {d\eta _{q}}{d\Theta }}{\bigg )}_{\epsilon _{q}}={\frac {\epsilon _{q}-{\overline {\epsilon }}_{q}}{\Theta ^{2}}}.$

(213)

That is, a diminution of the modulus will diminish the probability of all configurations for which the potential energy exceeds its average value in the ensemble. Again, in the ensemble having modulus $\Theta '$ and average energies ${\overline {\epsilon }}_{p}{}'$ and ${\overline {\epsilon }}_{q}{}'$, values of ${\overline {\epsilon }}_{q}$ sensibly less than ${\overline {\epsilon }}_{q}{}'$ will be so rare as to be practically negligible. They will be still more rare in an ensemble of greater modulus, since by the same equation an increase of the modulus will diminish the probability of configurations for which the potential energy is less than its average value in the ensemble. Therefore, for values of $\Theta$ between $\Theta '$ and $\Theta ''$, and of ${\overline {\epsilon }}_{p}$ between ${\overline {\epsilon }}_{p}{}'$ and ${\overline {\epsilon }}_{p}{}''$, the individual values of ${\overline {\epsilon }}_{q}$ will be practically limited to the interval between ${\overline {\epsilon }}_{q}{}'$ and ${\overline {\epsilon }}_{q}{}''$.

In the cases which remain to be considered, viz., when $d{\overline {\epsilon }}_{q}/d{\overline {\epsilon }}_{p}$ has very large values not confined to narrow limits, and consequently the differences of the mean potential energies in ensembles of different moduli are in general very large compared with the differences of the mean kinetic energies, it appears by (210) that the anomalies of mean square of potential energy, if not small in comparison with the mean kinetic energy, will yet in general be very small in comparison with differences of mean potential energy in ensembles having moderate differences of mean kinetic energy,—the exceptions being of the same character as described for the case when $d{\overline {\epsilon }}_{q}/d{\overline {\epsilon }}_{p}$ is not in general large.

It follows that to human experience and observation with respect to such an ensemble as we are considering, or with respect to systems which may be regarded as taken at random from such an ensemble, when the number of degrees of freedom is of such order of magnitude as the number of molecules in the bodies subject to our observation and experiment, $\epsilon -{\overline {\epsilon }}$, $\epsilon _{p}-{\overline {\epsilon }}_{p}$, $\epsilon _{q}-{\overline {\epsilon }}_{q}$ would be in general vanishing quantities, since such experience would not be wide enough to embrace the more considerable divergencies from the mean values, and such observation not nice enough to distinguish the ordinary divergencies. In other words, such ensembles would appear to human observation as ensembles of systems of uniform energy, and in which the potential and kinetic energies ( supposing that there were means of measuring these quantities separately) had each separately uniform values.^[4] Exceptions might occur when for particular values of the modulus the differential coefficient $d{\overline {\epsilon }}_{q}/d{\overline {\epsilon }}_{p}$ takes a very large value. To human observation the effect would be, that in ensembles in which $\Theta$ and ${\overline {\epsilon }}_{p}$ had certain critical values, ${\overline {\epsilon }}_{q}$ would be indeterminate within certain limits, viz., the values which would correspond to values of $\Theta$ and $\epsilon _{p}$ slightly less and slightly greater than the critical values. Such indeterminateness corresponds precisely to what we observe in experiments on the bodies which nature presents to us.^[5]

To obtain general formulae for the average values of powers of the energies, we may proceed as follows. If $h$ is any positive whole number, we have identically

$\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\epsilon ^{h}e^{-{\frac {\epsilon }{\Theta }}}\,dp_{1}\ldots dq_{n}=\Theta ^{2}{\frac {d}{d\Theta }}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}\epsilon ^{h-1}e^{-{\frac {\epsilon }{\Theta }}}\,dp_{1}\ldots dq_{n},$

(214)

i. e., by (108),

${\overline {\epsilon ^{h}}}e^{-{\frac {\psi }{\Theta }}}=\Theta ^{2}{\frac {d}{d\Theta }}{\bigg (}{\overline {\epsilon ^{h-1}}}e^{-{\frac {\psi }{\Theta }}}{\bigg )}.$

(215)

Hence

${\overline {\epsilon ^{h}}}e^{-{\frac {\psi }{\Theta }}}={\bigg (}\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}^{h}e^{-{\frac {\psi }{\Theta }}},$

(216)

and

${\overline {\epsilon ^{h}}}=e^{\frac {\psi }{\Theta }}{\bigg (}\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}^{h}e^{-{\frac {\psi }{\Theta }}}.$

(217)

For $h=1$, this gives

${\overline {\epsilon }}=-\Theta ^{2}{\frac {d}{d\Theta }}{\bigg (}{\frac {\psi }{\Theta }}{\bigg )}$

(218)

which agrees with (191).
From (215) we have also

${\overline {\epsilon ^{h}}}={\overline {\epsilon }}{\overline {\epsilon ^{h-1}}}+\Theta ^{2}{\frac {d{\overline {\epsilon ^{h-1}}}}{d\Theta }}={\bigg (}{\overline {\epsilon }}+\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}\epsilon ^{h-1},$

(219)

${\overline {\epsilon ^{h}}}={\bigg (}{\overline {\epsilon }}+\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}^{h-1}{\overline {\epsilon }}.$

(220)

In like manner from the identical equation

$\mathop {\int \ldots \int } _{\rm {config.}}^{\rm {all}}\epsilon _{q}{}^{h}e^{-{\frac {\epsilon _{q}}{\Theta }}}\Delta _{\dot {q}}{}^{\frac {1}{2}}\,dq_{1}\ldots dq_{n}=\Theta ^{2}{\frac {d}{d\Theta }}\mathop {\int \ldots \int } _{\rm {config.}}^{\rm {all}}\epsilon _{q}{}^{h-1}e^{-{\frac {\epsilon _{q}}{\Theta }}}\Delta _{\dot {q}}{}^{\frac {1}{2}}\,dq_{1}\ldots dq_{n},$

(221)

we get

${\overline {\epsilon _{q}{}^{h}}}=e^{\frac {\psi _{q}}{\Theta }}{\bigg (}\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}^{h}e^{-{\frac {\psi _{q}}{\Theta }}}.,$

(222)

and

${\overline {\epsilon _{q}{}^{h}}}={\bigg (}{\overline {\epsilon }}_{q}+\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}^{h-1}{\overline {\epsilon }}_{q}.$

(223)

With respect to the kinetic energy similar equations will hold for averages taken for any particular configuration, or for the whole ensemble. But since

${\overline {\epsilon }}_{p}={\frac {n}{2}}\Theta ,$

the equation

${\overline {\epsilon _{p}{}^{h}}}={\bigg (}{\overline {\epsilon }}_{p}+\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}{\overline {\epsilon _{p}{}^{h-1}}}$

(224)

reduces to

${\overline {\epsilon _{p}{}^{h}}}={\bigg (}{\frac {n}{2}}\Theta +{}^{2}\Theta {\frac {d}{d\Theta }}{\bigg )}{\overline {\epsilon _{p}{}^{h-1}}}={\frac {n}{2}}{\bigg (}{\frac {n}{2}}\Theta +\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}^{h-1}\Theta .$

(225)

We have therefore

${\overline {\epsilon _{p}{}^{2}}}={\bigg (}{\frac {n}{2}}+1{\bigg )}{\frac {n}{2}}\Theta ^{2}.$

(226)

${\overline {\epsilon _{p}{}^{3}}}={\bigg (}{\frac {n}{2}}+2{\bigg )}{\bigg (}{\frac {n}{2}}+1{\bigg )}{\frac {n}{2}}\Theta ^{2}.$

(227)

${\overline {\epsilon _{p}{}^{h}}}={\frac {\Gamma ({\tfrac {1}{2}}n+h)}{\Gamma ({\tfrac {1}{2}}n)}}\Theta ^{h}.$

[6](228)

The average values of the powers of the anomalies of the energies are perhaps most easily found as follows. We have identically, since ${\overline {\epsilon }}$ is a function of $\Theta$, while $\epsilon$ is a function of the $p$'s and $q$'s,

${\begin{aligned}&\Theta ^{2}{\frac {d}{d\Theta }}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}(\epsilon -{\overline {\epsilon }})^{h}e^{-{\frac {\epsilon }{\Theta }}}\,dp_{1},\ldots dq_{n}=\\&\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\bigg [}\epsilon (\epsilon -{\overline {\epsilon }})^{h}-h(\epsilon -{\overline {\epsilon }})^{h-1}\Theta ^{2}{\frac {d{\overline {\epsilon }}}{d\Theta }}{\bigg ]}e^{-{\frac {\epsilon }{\Theta }}}\,dp_{1},\ldots dq_{n}\end{aligned}}$

(229)

i. e., by (108),

$\Theta ^{2}{\frac {d}{d\Theta }}{\bigg [}{\overline {(\epsilon -{\overline {\epsilon }})^{h}}}e^{-{\frac {\psi }{\Theta }}}{\bigg ]}={\bigg [}{\overline {\epsilon (\epsilon -{\overline {\epsilon }})^{h}}}-h{\overline {(\epsilon -{\overline {\epsilon }})^{h-1}}}\Theta ^{2}{\frac {d{\overline {\epsilon }}}{d\Theta }}{\bigg ]}e^{-{\frac {\psi }{\Theta }}},$

(230)

or since by (218)

$\Theta ^{2}{\frac {d}{d\Theta }}e^{-{\frac {\psi }{\Theta }}}={\overline {\epsilon }}e^{-{\frac {\psi }{\Theta }}},$

$\Theta ^{2}{\frac {d}{d\Theta }}{\overline {(\epsilon -{\overline {\epsilon }})^{h}}}+{\overline {(\epsilon -{\overline {\epsilon }})^{h}}}{\overline {\epsilon }}={\overline {\epsilon (\epsilon -{\overline {\epsilon }})^{h}}}-h{\overline {(\epsilon -{\overline {\epsilon }})^{h-1}}}\Theta ^{2}{\frac {d{\overline {\epsilon }}}{d\Theta }},$

${\overline {(\epsilon -{\overline {\epsilon }})^{h+1}}}=\Theta ^{2}{\frac {d}{d\Theta }}{\overline {(\epsilon -{\overline {\epsilon }})^{h}}}+h{\overline {(\epsilon -{\overline {\epsilon }})^{h-1}}}\Theta ^{2}{\frac {d{\overline {\epsilon }}}{d\Theta }}.$

(231)

In precisely the same way we may obtain for the potential energy

${\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{h+1}}}=\Theta ^{2}{\frac {d}{d\Theta }}{\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{h}}}+h{\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{h-1}}}\Theta ^{2}{\frac {d{\overline {\epsilon }}_{q}}{d\Theta }}.$

(232)

By successive applications of (231) we obtain

${\begin{aligned}{\overline {(\epsilon -{\overline {\epsilon }})^{2}}}&=D{\overline {\epsilon }}\\{\overline {(\epsilon -{\overline {\epsilon }})^{3}}}&=D^{2}{\overline {\epsilon }}\\{\overline {(\epsilon -{\overline {\epsilon }})^{4}}}&=D^{3}{\overline {\epsilon }}+3(D{\overline {\epsilon }})^{2}\\{\overline {(\epsilon -{\overline {\epsilon }})^{5}}}&=D^{4}{\overline {\epsilon }}+10D{\overline {\epsilon }}D^{2}{\overline {\epsilon }}\\{\overline {(\epsilon -{\overline {\epsilon }})^{6}}}&=D^{5}{\overline {\epsilon }}+15D{\overline {\epsilon }}D^{3}{\overline {\epsilon }}+10(D^{2}{\overline {\epsilon }})^{2}+15(D{\overline {\epsilon }})^{3}\mathrm {etc.} \end{aligned}}$

where $D$ represents the operator $\Theta ^{2}\,d/d\Theta$. Similar expressions relating to the potential energy may be derived from (232).

For the kinetic energy we may write similar equations in which the averages may be taken either for a single configuration or for the whole ensemble. But since

${\frac {d{\overline {\epsilon }}_{p}}{d\Theta }}={\frac {n}{2}}$

the general formula reduces to

${\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{h+1}}}=\Theta ^{2}{\frac {d}{d\Theta }}{\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{h}}}+{\tfrac {1}{2}}nh\Theta ^{2}{\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{h-1}}}$

(233)

or

${\frac {\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{h+1}}}{{\overline {\epsilon }}_{p}{}^{h+1}}}={\frac {2\Theta }{n}}{\frac {d}{d\Theta }}{\frac {\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{h}}}{{\overline {\epsilon }}_{p}{}^{h}}}+{\frac {2h}{n}}{\frac {\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{h}}}{{\overline {\epsilon }}_{p}{}^{h}}}+{\frac {2h}{n}}{\frac {\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{h-1}}}{{\overline {\epsilon }}_{p}{}^{h-1}}}$

(234)

But since identically

${\frac {\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{0}}}{{\overline {\epsilon }}_{p}{}^{0}}}=1,\quad {\frac {\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{1}}}{{\overline {\epsilon }}_{p}{}^{1}}}=0$

the value of the corresponding expression for any index will be independent of $\Theta$ and the formula reduces to

${\overline {{\bigg (}{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{h+1}}}={\frac {2h}{n}}{\overline {{\bigg (}{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{h}}}+{\frac {2h}{n}}{\overline {{\bigg (}{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{h-1}}}$

(235)

we have therefore

${\overline {{\bigg (}{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{0}}}=1,\quad {\overline {{\bigg (}{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{3}}}={\frac {8}{n^{2}}},$

${\overline {{\bigg (}{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{1}}}=0,\quad {\overline {{\bigg (}{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{4}}}={\frac {48}{n^{3}}}+{\frac {12}{n^{2}}},$

${\overline {{\bigg (}{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{2}}}={\frac {2}{n}},\quad \mathrm {etc.}$[7]

It will be observed that when $\psi$ or ${\overline {\epsilon }}$ is given as function of $\Theta$, all averages of the form ${\overline {\epsilon ^{h}}}$ or ${\overline {(\epsilon -{\overline {\epsilon }})^{h}}}$ are thereby

determined. So also if $\psi _{q}$ or ${\overline {\epsilon }}_{q}$ is given as a function of $\Theta$, all averages of the form ${\overline {\epsilon _{q}{}^{h}}}$ or ${\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{h})}}$ are determined. But

${\overline {\epsilon }}_{q}={\overline {\epsilon }}-{\tfrac {1}{2}}n\Theta .$

Therefore if any one of the quantities $\psi$, $\psi _{q}$, ${\overline {\epsilon }}$, ${\overline {\epsilon }}_{q}$ is known as a function of $\Theta$, and $n$ is also known, all averages of any of the forms mentioned are thereby determined as functions of the same variable. In any case all averages of the form

${\overline {{\bigg (}{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{h}}}$

are known in terms of $n$ alone, and have the same value whether taken for the whole ensemble or limited to any particular configuration.

If we differentiate the equation

$\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}=1$

(236)

with respect to $a_{1}$, and multiply by $\Theta$, we have

$\int \ldots \int {\bigg [}{\frac {d\psi }{da_{1}}}-{\frac {d\epsilon }{da_{1}}}{\bigg ]}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}=0.$

(237)

Differentiating again, with respect to $a_{1}$, with respect to $a_{2}$, and with respect to $\Theta$, we have

$\int \ldots \int {\bigg [}{\frac {d^{2}\psi }{da_{1}{}^{2}}}-{\frac {d^{2}\epsilon }{da_{1}{}^{2}}}+{\frac {1}{\Theta }}{\bigg (}{\frac {d\psi }{da_{1}}}-{\frac {d\epsilon }{da_{1}}}{\bigg )}^{2}{\bigg ]}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}=0,$

(238)

${\begin{aligned}\int \ldots \int {\bigg [}{\frac {d^{2}\psi }{da_{1}\,da_{2}}}-{\frac {d^{2}\epsilon }{da_{1}\,da_{2}}}&+{\frac {1}{\Theta }}{\bigg (}{\frac {d\psi }{da_{1}}}-{\frac {d\epsilon }{da_{1}}}{\bigg )}{\bigg (}{\frac {d\psi }{da_{2}}}-{\frac {d\epsilon }{da_{2}}}{\bigg )}{\bigg ]}\\&\qquad \qquad \qquad e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}=0,\end{aligned}}$

(239)

${\begin{aligned}\int \ldots \int {\bigg [}{\frac {d^{2}\psi }{da_{1}\,d\Theta }}+{\bigg (}{\frac {d\psi }{da_{1}}}-{\frac {d\epsilon }{da_{1}}}{\bigg )}&{\bigg (}{\frac {1}{\Theta }}{\frac {d\psi }{d\Theta }}-{\frac {\psi -\epsilon }{\Theta ^{2}}}{\bigg )}{\bigg ]}\\&\qquad e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}=0.\end{aligned}}$

(240)

The multiple integrals in the last four equations represent the average values of the expressions In the brackets, which we may therefore set equal to zero. The first gives

${\frac {d\psi }{da_{1}}}={\overline {\frac {d\epsilon }{da_{1}}}}=-{\overline {A}}_{1},$

(241)

as already obtained. With this relation and (191) we get from the other equations

${\overline {(A_{1}-{\overline {A}}_{1})^{2}}}=\Theta {\bigg (}{\overline {\frac {d^{2}\epsilon }{da_{1}{}^{2}}}}-{\frac {d^{2}\psi }{da_{1}{}^{2}}}{\bigg )}=\Theta {\bigg (}{\frac {d{\overline {A}}_{1}}{da_{1}}}-{\overline {\frac {dA_{1}}{da_{1}}}}{\bigg )}$

(242)

${\begin{aligned}{\overline {(A_{1}-{\overline {A}}_{1})(A_{2}-{\overline {A}}_{2})}}&=\Theta {\bigg (}{\overline {\frac {d^{2}\epsilon }{da_{1}\,da_{2}}}}-{\frac {d^{2}\psi }{da_{1}\,da_{2}}}{\bigg )}\\&=\Theta {\bigg (}{\frac {d{\overline {A}}_{1}}{da_{2}}}-{\overline {\frac {dA_{1}}{da_{2}}}}{\bigg )}=\Theta {\bigg (}{\frac {d{\overline {A}}_{2}}{da_{1}}}-{\overline {\frac {dA_{2}}{da_{1}}}}{\bigg )}\end{aligned}}$

(243)

${\overline {(A_{1}-{\overline {A}}_{1})(\epsilon -{\overline {\epsilon }})}}=-\Theta ^{2}{\frac {d^{2}\psi }{da_{1}\,d\Theta }}=\Theta ^{2}{\frac {d{\overline {A}}_{1}}{d\Theta }}=-\Theta ^{2}{\frac {d{\overline {\eta }}}{da_{1}}}.$

We may add for comparison equation (205), which might be derived from (236) by differentiating twice with respect to $\Theta$:

${\overline {(\epsilon -{\overline {\epsilon }})^{2}}}=-\Theta ^{2}{\frac {d^{2}\psi }{d\Theta ^{2}}}=\Theta ^{2}{\frac {d{\overline {\epsilon }}}{d\Theta }}.$

(244)

The two last equations give

${\overline {(A_{1}-{\overline {A}}_{1})(\epsilon -{\overline {\epsilon }})}}={\frac {d{\overline {A}}}{d{\overline {\epsilon }}}}{\overline {(\epsilon -{\overline {\epsilon }})^{2}}}.$

(245)

If $\psi$ or ${\overline {\epsilon }}$ is known as function of $\Theta$, $a_{1}$, $a_{2}$, etc., ${\overline {(\epsilon -{\overline {\epsilon }})^{2}}}$ may be obtained by differentiation as function of the same variables. And if $\psi$, or ${\overline {A}}_{1}$, or ${\overline {\eta }}$ is known as function of $\Theta$, $a_{1}$, etc., ${\overline {(A_{1}-{\overline {A}}_{1})(\epsilon -{\overline {\epsilon }})}}$ may be obtained by differentiation. But ${\overline {(A_{1}-{\overline {A}}_{1})^{2}}}$ and ${\overline {(A_{1}-{\overline {A}}_{1})(A_{2}-{\overline {A}}_{2})}}$ cannot be obtained in any similar manner. We have seen that ${\overline {(\epsilon -{\overline {\epsilon }})^{2}}}$ is in general a vanishing quantity for very great values of $n$, which we may regard as contained implicitly in $\Theta$ as a divisor. The same is true of ${\overline {(A_{1}-{\overline {A}}_{1})(\epsilon -{\overline {\epsilon }})}}$. It does not appear that we can assert the same of ${\overline {(A_{1}-{\overline {A}}_{1})^{2}}}$ or ${\overline {(A_{1}-{\overline {A}}_{1})(A_{2}-{\overline {A}}_{2})}}$, since $d^{2}\epsilon /da_{1}{}^{2}$ may be very great. The quantities $d^{2}\epsilon /da_{1}{}^{2}$ and $d^{2}\psi /da_{1}{}^{2}$ belong to the class called elasticities. The former expression represents an elasticity measured under the condition that while $a_{1}$ is varied the internal coördinates $q_{1},\ldots q_{n}$ all remain fixed. The latter is an elasticity measured under the condition that when $a_{1}$ is varied the ensemble remains canonically distributed within the same modulus. This corresponds to an elasticity in physics measured under the condition of constant temperature. It is evident that the former is greater than the latter, and it may be enormously greater.

The divergences of the force $A_{1}$ from its average value are due in part to the differences of energy in the systems of the ensemble, and in part to the differences in the value of the forces which exist in systems of the same energy. If we write ${\overline {A_{1}}}|_{\epsilon }$ for the average value of $A_{1}$ in systems of the ensemble which have any same energy, it will be determined by the equation

${\overline {A_{1}}}|_{\epsilon }={\frac {\int \ldots \int -{\frac {d\epsilon }{da_{1}}}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}}{\int \ldots \int e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}}}$

(246)

where the limits of integration in both multiple integrals are two values of the energy which differ infinitely little, say $\epsilon$ and $\epsilon +d\epsilon$. This will make the factor $e^{\frac {\psi -\epsilon }{\Theta }}$ constant within the limits of integration, and it may be cancelled in the numerator and denominator, leaving

${\overline {A_{1}}}|_{\epsilon }={\frac {\int \ldots \int -{\frac {d\epsilon }{da_{1}}}\,dp_{1}\ldots dq_{n}}{\int \ldots \int \,dp_{1}\ldots dq_{n}}}$

(247)

where the integrals as before are to be taken between $\epsilon$ and $\epsilon +d\epsilon$. ${\overline {A_{1}}}|_{\epsilon }$ is therefore independent of $\Theta$, being a function of the energy and the external coördinates.

Now we have identically

$A_{1}-{\overline {A}}_{1}=(A_{1}-{\overline {A_{1}}}|_{\epsilon })+({\overline {A_{1}}}|_{\epsilon }-{\overline {A}}_{1}),$

where $A_{1}-{\overline {A_{1}}}|_{\epsilon }$ denotes the excess of the force (tending to increase $a_{1}$ exerted by any system above the average of such forces for systems of the same energy. Accordingly,

${\overline {(A_{1}-{\overline {A}}_{1})^{2}}}={\overline {(A_{1}-{\overline {A_{1}}}|_{\epsilon })^{2}}}+2{\overline {(A_{1}-{\overline {A_{1}}}|_{\epsilon })({\overline {A_{1}}}|_{\epsilon }-{\overline {A}}_{1})}}+{\overline {({\overline {A_{1}}}|_{\epsilon }-{\overline {A}}_{1})^{2}}}.$

But the average value of $(A_{1}-{\overline {A_{1}}}|_{\epsilon })({\overline {A_{1}}}|_{\epsilon }-{\overline {A}}_{1})$ for systems of the ensemble which have the same energy is zero, since for such systems the second factor is constant. Therefore the average for the whole ensemble is zero, and

${\overline {(A_{1}-{\overline {A}}_{1})^{2}}}={\overline {(A_{1}-{\overline {A_{1}}}|_{\epsilon })^{2}}}+{\overline {({\overline {A_{1}}}|_{\epsilon }-{\overline {A}}_{1})^{2}}}.$

(248)

In the same way it may be shown that

${\overline {(A_{1}-{\overline {A}}_{1})(\epsilon -{\overline {\epsilon }})}}={\overline {({\overline {A_{1}}}|_{\epsilon }-{\overline {A}}_{1})(\epsilon -{\overline {\epsilon }})}}.$

(249)

It is evident that in ensembles in which the anomalies of energy $\epsilon -{\overline {\epsilon }}$ may be regarded as insensible the same will be true of the quantities represented by ${\overline {A_{1}}}|_{\epsilon }-{\overline {A}}_{1}$.

The properties of quantities of the form ${\overline {A_{1}}}|_{\epsilon }$ will be farther considered in Chapter X, which will be devoted to ensembles of constant energy.

It may not be without interest to consider some general formulae relating to averages in a canonical ensemble, which embrace many of the results which have been given in this chapter.

Let $u$ be any function of the internal and external coördinates with the momenta and modulus. We have by definition

${\overline {u}}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}ue^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}$

(250)

If we differentiate with respect to $\Theta$, we have

${\frac {d{\overline {u}}}{d\Theta }}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\bigg (}{\frac {du}{d\Theta }}-{\frac {u}{\Theta ^{2}}}(\psi -\epsilon )+{\frac {u}{\Theta }}{\frac {d\psi }{d\Theta }}{\bigg )}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n},$

or

${\frac {d{\overline {u}}}{d\Theta }}={\overline {\frac {du}{d\Theta }}}-{\overline {\frac {u(\psi -\epsilon )}{\Theta ^{2}}}}+{\frac {\overline {u}}{\Theta }}{\frac {d\psi }{d\Theta }}.$

(251)

Setting $u=1$ in this equation, we get

${\frac {d\psi }{d\Theta }}={\frac {\psi -{\overline {\epsilon }}}{\Theta }},$

and substituting this value, we have

${\frac {\overline {u}}{d\Theta }}={\overline {\frac {du}{d\Theta }}}+{\frac {\overline {u\epsilon }}{\Theta ^{2}}}-{\frac {{\overline {u}}\,{\overline {\epsilon }}}{\Theta ^{2}}}$

or

$\Theta ^{2}{\frac {\overline {u}}{d\Theta }}-\Theta ^{2}{\overline {\frac {du}{d\Theta }}}={\overline {u\epsilon }}-{\overline {u}}\,{\overline {\epsilon }}={\overline {(u-{\overline {u}})(\epsilon -{\overline {\epsilon }})}}.$

(252)

If we differentiate equation (250) with respect to $a$ (which may represent any of the external coördinates), and write $A$ for the force $-{\frac {d\epsilon }{dA}}$, we get

${\frac {\overline {u}}{da}}=\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\bigg (}{\frac {du}{da}}+{\frac {u}{\Theta }}{\frac {d\psi }{da}}+{\frac {u}{\Theta }}A{\bigg )}e^{\frac {\psi -\epsilon }{\Theta }}\,dp_{1}\ldots dq_{n}$

or

${\frac {\overline {u}}{da}}={\overline {\frac {du}{da}}}+{\frac {\overline {u}}{\Theta }}{\frac {d\psi }{da}}+{\frac {\overline {uA}}{\Theta }}$

(253)

Setting $u=1$ in this equation, we get

${\frac {d\psi }{da}}=-{\overline {A}}.$

Substituting this value, we have

${\frac {\overline {u}}{da}}={\overline {\frac {du}{da}}}+{\frac {\overline {uA}}{\Theta }}-{\frac {{\overline {u}}\,{\overline {A}}}{\Theta }},$

(254)

or

$\Theta {\frac {\overline {u}}{da}}-\Theta {\overline {\frac {du}{da}}}={\overline {uA}}-{\overline {u}}\,{\overline {A}}={\overline {(u-{\overline {u}})(A-{\overline {A}})}}.$

(255)

Repeated applications of the principles expressed by equations (252) and (255) are perhaps best made in the particular cases. Yet we may write (252) in this form

${\overline {(\epsilon +D)(u-{\overline {u}})}}=0,$

(256)

where $D$ represents the operator $\Theta ^{2}\,d/d\Theta$.
Hence

${\overline {(\epsilon +D)^{h}(u-{\overline {u}})}}=0,$

(257)

where $h$ is any positive whole number. It will be observed, that since $\epsilon$ is not function of $\Theta$, $(\epsilon +D)^{h}$ may be expanded by the binomial theorem. Or, we may write

${\overline {(\epsilon +D)u}}=({\overline {\epsilon }}+D){\overline {u}},$

(258)

whence

${\overline {(\epsilon +D)^{h}u}}=({\overline {\epsilon }}+D)^{h}{\overline {u}}.$

(259)

But the operator $({\overline {\epsilon }}+D)^{h}$, although in some respects more simple than the operator without the average sign on the $\epsilon$, cannot be expanded by the binomial theorem, since ${\overline {\epsilon }}$ is a function of $\Theta$ with the external coördinates.

So from equation (254) we have

${\overline {{\bigg (}{\frac {A}{\Theta }}+{\frac {d}{da}}{\bigg )}(u-{\overline {u}})}}=0,$

(260)

whence

${\overline {{\bigg (}{\frac {A}{\Theta }}+{\frac {d}{da}}{\bigg )}^{h}(u-{\overline {u}})}}=0;$

(261)

and

${\overline {{\bigg (}{\frac {A}{\Theta }}+{\frac {d}{da}}{\bigg )}u}}={\bigg (}{\frac {\overline {A}}{\Theta }}+{\frac {d}{da}}{\bigg )}{\overline {u}},$

(262)

whence

${\overline {{\bigg (}{\frac {A}{\Theta }}+{\frac {d}{da}}{\bigg )}^{h}u}}={\bigg (}{\frac {\overline {A}}{\Theta }}+{\frac {d}{da}}{\bigg )}^{h}{\overline {u}}.$

(263)

The binomial theorem cannot be applied to these operators.

Again, if we now distinguish, as usual, the several external coördinates by suffixes, we may apply successively to the expression $u-{\overline {u}}$ any or all of the operators

$\epsilon +\Theta ^{2}{\frac {d}{d\Theta }},\quad A_{1}+\Theta {\frac {d}{da_{1}}},\quad A_{2}+\Theta {\frac {d}{da_{2}}},\quad \mathrm {etc.}$

(264)

as many times as we choose, and in any order, the average value of the result will be zero. Or, if we apply the same operators to $u$, and finally take the average value, it will be the same as the value obtained by writing the sign of average separately as $u$, and on $\epsilon$, $A_{1}$, $A_{2}$, etc., in all the operators.

If $u$ is independent of the momenta, formulae similar to the preceding, but having $\epsilon _{q}$ in place of $\epsilon$, may be derived from equation (179).

---

1. Pogg. Ann., Bd. CXVI, S. 73, (1862); ibid., Bd. CXXV, S. 353, (1865). See also Boltzmann, Sitzb. der Wiener Akad., Bd. LXIII, S. 728, (1871).
2. Verwandlungswerth des Wärmeinhaltes.
3. In the case discussed in the note on page 54, in which the potential energy is a quadratic function of the $q$'s, and $\Delta _{\dot {q}}$ independent of the $q$'s, we should get for the potential energy

   ${\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{2}}}={\frac {1}{2}}n\Theta ^{2},$

   and for the total energy

   ${\overline {(\epsilon -{\overline {\epsilon }})^{2}}}=n\Theta ^{2}.$

   We may also write in this case,

   ${\frac {\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{2}}}{({\overline {\epsilon }}_{q}-\epsilon _{a})^{2}}}={\frac {2}{n}},$

   ${\frac {\overline {(\epsilon -{\overline {\epsilon }})^{2}}}{({\overline {\epsilon }}-\epsilon _{a})^{2}}}={\frac {1}{n}}.$

4. This implies that the kinetic and potential energies of individual systems would each separately have values sensibly constant in time.
5. As an example, we may take a system consisting of a fluid in a cylinder under a weighted piston, with a vacuum between the piston and the top of the cylinder, which is closed. The weighted piston is to be regarded as a part of the system. (This is formally necessary in order to satisfy the condition of the invariability of the external coördinates.) It is evident that at a certain temperature, viz., when the pressure of saturated vapor balances the weight of the piston, there is an indeterminateness in the values of the potential and total energies as functions of the temperature.
6. In the case discussed in the note on page 54 we may easily get

   ${\overline {(\epsilon _{q}-\epsilon _{a})^{h}}}={\bigg (}{\overline {\epsilon }}_{q}-\epsilon _{a}+\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}{\overline {(\epsilon _{q}-\epsilon _{a})^{h-1}}},$

   which, with

   ${\overline {\epsilon }}_{q}-\epsilon _{a}={\frac {n}{2}}\Theta ,$

   gives

   ${\overline {(\epsilon _{q}-\epsilon _{a})^{h}}}={\bigg (}{\frac {n}{2}}\Theta +\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}{\overline {(\epsilon _{q}-\epsilon _{a})^{h-1}}}={\frac {n}{2}}{\bigg (}{\frac {n}{2}}\Theta +\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}^{h-1}\Theta .$

   Hence

   ${\overline {(\epsilon _{q}-\epsilon _{a})^{h}}}={\overline {\epsilon _{p}{}^{h}}}.$

   Again

   ${\overline {(\epsilon -\epsilon _{a})^{h}}}={\bigg (}{\overline {\epsilon }}-\epsilon _{a}+\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}{\overline {(\epsilon -\epsilon _{a})^{h-1}}},$

   which with

   ${\overline {\epsilon }}-\epsilon _{a}=n\Theta$

   gives

   ${\overline {(\epsilon -\epsilon _{a})^{h}}}={\bigg (}n\Theta +\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}{\overline {(\epsilon -\epsilon _{a})^{h-1}}}=n{\bigg (}n\Theta +\Theta ^{2}{\frac {d}{d\Theta }}{\bigg )}^{h-1}\Theta ,$

   hence

   ${\overline {(\epsilon -\epsilon _{a})^{h}}}={\frac {\Gamma (n+h)}{\Gamma (n)}}\Theta ^{h}.$

7. In the case discussed in the preceding foot-notes we get easily

   ${\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{h}}}={\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{h}}},$

   and

   ${\overline {{\bigg (}{\frac {\epsilon _{q}-{\overline {\epsilon }}_{q}}{{\overline {\epsilon }}_{q}-\epsilon _{a}}}{\bigg )}^{h}}}={\overline {{\bigg (}{\frac {\epsilon _{p}-{\overline {\epsilon }}_{p}}{{\overline {\epsilon }}_{p}}}{\bigg )}^{h}}}.$

   For the total energy we have in this case

   ${\overline {{\bigg (}{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{h+1}}}={\frac {h}{n}}{\overline {{\bigg (}{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{h}}}+{\overline {{\frac {h}{n}}{\bigg (}{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{h-1}}}.$

   ${\overline {{\bigg (}{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{2}}}={\frac {1}{n}}.\quad {\overline {{\bigg (}{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{4}}}={\frac {3}{n^{2}}}+{\frac {6}{n^{3}}},$

   ${\overline {{\bigg (}{\frac {\epsilon -{\overline {\epsilon }}}{{\overline {\epsilon }}-\epsilon _{a}}}{\bigg )}^{3}}}={\frac {2}{n^{2}}},\quad \mathrm {etc.}$